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So for whatever reason I just can't wrap my head around this, I know I am doing it wrong.

Question: Two cards are chosen from a pack of cards without replacement. Are the following events independent? (i)the first card is a heart, (ii)the second card is a picture card.

For these events to be independent $P(i \cap ii) = P(i)P(ii)$ and $P(i|ii) = P(i)$ and $P(ii|i) = P(ii)$.

In this case $P(i)={13\over 52}$ and $P(ii)= {12\over 51}$

So $P(i|ii) = {P(i\cap ii)\over P(ii)} = {{3\over 52} \over {12\over 51}} = {51\over 208} \not= {13\over 52}$

According to the book, these events are independent, but I get that they are not. I need help understanding this. What's wrong with my approach?

Nolohice
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2 Answers2

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$P(ii)=\frac {12}{52}$ because it is not conditioned on what the first card was. $P(ii|i)$ is still $\frac 3{13}$-If you go through the chance that the first card is or is not a picture card you will find that. Yes, they are independent because the density of picture cards among the hearts is the same as the density of picture cards among the rest of the deck.

Ross Millikan
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  • So the error is in assuming the cards are drawn in succession rather than thinking of a scenario where the first card is a heart and the second is a picture card, in a deck before any type of drawing is made? – Nolohice Mar 03 '15 at 20:36
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    Your value of $P(ii)$ assumes that the first card drawn is a non-picture card. That is where the denominator of $51$ comes from. We are not given that. – Ross Millikan Mar 03 '15 at 20:43
  • Could I instead do ${{3\over 52} * {10\over 52}\over {12\over 51} * {11\over 51}}$ I think that might make more intuitive since to me – Nolohice Mar 03 '15 at 21:01
  • I do not understand what that is supposed to represent. If you want $P(ii|i)$ you can do $\frac 3{13}$ (heart is a picture)$\frac {11}{51}$ (second card is a picture) $+\frac {10}{13}$ (heart is not a picture)$\frac {12}{51}$ (second card is a picture) – Ross Millikan Mar 03 '15 at 21:10
  • $3\over 52$ is the event that the first card is a heart & picture. $10\over 52$ is the event that the first card is a heart and not a picture. $12\over 51$ is the event that the second card is a picture card, given that the first card was not a picture card. $11\over 51$ is the event that the second card is a picture card given that the first card was also a picture card.

    But I think I see what you're saying now. Thank you!

     – Nolohice Mar 03 '15 at 21:19
  • You shouldn't be multiplying the chances that the first card is a heart and picture and the chance that the first card is heart and not a picture-they are exclusive events. You should be multiplying things about the first card and the second card, as I did in my last comment. – Ross Millikan Mar 03 '15 at 21:22
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$P(ii) = 12/52$ as there are 12 picture cards in the pack of 52 and any card can be second from the top with equal probability.

Graham Kemp
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